Problem 12 - A Graphical Approach to College Algebra

Question

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\left(\frac{1}{3}\right)^{\sqrt{6}}$$

Step-by-Step Solution

Answer

\( \left(\frac{1}{3}\right)^{\sqrt{6}} \approx 0.1464057 \).

Step 1: Understand the Expression

The expression we need to evaluate is \( \left(\frac{1}{3}\right)^{\sqrt{6}} \). This means we are raising \( \frac{1}{3} \) to the power of the square root of 6.

Step 2: Simplify the Power

Calculate \( \sqrt{6} \). Using a calculator, \( \sqrt{6} \approx 2.4494897 \).

Step 3: Compute the Power

Now, raise \( \frac{1}{3} \) to the power of the approximate value of \( \sqrt{6} \). Use a calculator to find the value of \( \left(\frac{1}{3}\right)^{2.4494897} \).

Step 4: Obtain the Approximate Value

Using the calculator, \( \left(\frac{1}{3}\right)^{2.4494897} \approx 0.1464057 \). Ensure that the calculator is set to display the maximum number of decimal places possible.

Key Concepts

Calculator Usage

When dealing with complex mathematical operations like exponentiation, calculators become invaluable. Most scientific calculators can handle powers and roots efficiently.
To use your calculator for an expression like \( \left(\frac{1}{3}\right)^{\sqrt{6}} \), follow these steps:

• First, calculate the square root part. Enter 6 and use the square root function. Your calculator should display approximately 2.4494897.
• Next, raise \( \frac{1}{3} \) to this power. Enter \( \frac{1}{3} \) or 0.3333 (if your calculator requires a decimal form), press the power button, and enter 2.4494897.
• Ensure your calculator is set to provide the maximum number of decimal places possible to retain accuracy.

Understanding how to correctly input these values is crucial for getting the right answer. Always double-check your entries to avoid errors.

Decimal Approximation

Decimals can make or break mathematical calculations, especially when accuracy is key. Decimal approximation is a method of presenting numbers in a simpler form while maintaining an acceptable degree of precision.
When you compute \( \left(\frac{1}{3}\right)^{2.4494897} \), the exact result may not be easily expressible, so calculators give a decimal approximation. In this case, the result is approximately 0.1464057.
Here are some tips on handling decimals:

• Understand that more decimal places mean more accuracy, so use them wisely in calculations.
• Be aware of your calculator settings to ensure maximum decimal place display, which helps in keeping computations precise.
• Use decimals to compare or estimate where exact figures are not necessary but maintain as many places as needed for precision.

Approximating decimals helps simplify complex numbers and keeps your work manageable.

Square Root Computation

Square root computation is an essential part of mathematics, and it's needed when dealing with expressions like \( \left(\frac{1}{3}\right)^{\sqrt{6}} \). To compute the square root of 6 accurately, you rely on calculators for exactness and ease.
Here's a straightforward way to think about it:

• The square root of a number is a value that, when multiplied by itself, gives the original number.
• For \( \sqrt{6} \), you find a number which when squared equals 6. Calculators quickly compute this, showing approximately 2.4494897.
• Knowing how to find square roots helps in solving more complex problems involving powers and roots.

Understanding how to compute and use square roots allows you to handle expressions with confidence and accuracy. It's a key skill in many areas of math and science.